Dear All, I'm wondering whether the category of ALL group actions is locally Cartesian closed. This is NOT the functor category [G,Set] for some category G with one object, since we allow G to vary. To be more specific the category is as follows. - The objects are pairs (G,X), where G is a group and X is a G-Set. - A morphism (G,X) -> (G', X') is given by a pair (h,f), where h:G->G' is a group homomorphism and f: X -> X' is a function (a morphism in Set) such that for all g in G, x in X h(g) * f(x) = f(g * x) where * on the left denotes the group action of G' on X' and * on the right denotes the group action of G on X. All the best, Tim -- Timothy Revell, Department of Computer and Information Sciences, University of Strathclyde. The University of Strathclyde is a charitable body, registered in Scotland, with registration number SC015263. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Tim, Write Act for the category you mention. Then Act/(G,0) is clearly equivalent to Grp/G. Since Grp/G is not cartesian closed, neither is Act/(G,0). Regards, Steve Lack. On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:
Dear All,
I'm wondering whether the category of ALL group actions is locally Cartesian closed. This is NOT the functor category [G,Set] for some category G with one object, since we allow G to vary. To be more specific the category is as follows.
- The objects are pairs (G,X), where G is a group and X is a G-Set.
- A morphism (G,X) -> (G', X') is given by a pair (h,f), where h:G->G' is a group homomorphism and f: X -> X' is a function (a morphism in Set) such that for all g in G, x in X
h(g) * f(x) = f(g * x)
where * on the left denotes the group action of G' on X' and * on the right denotes the group action of G on X.
All the best, Tim
-- Timothy Revell, Department of Computer and Information Sciences, University of Strathclyde. The University of Strathclyde is a charitable body, registered in Scotland, with registration number SC015263.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:
I'm wondering whether the category of ALL group actions is locally Cartesian closed.
This is what I answered Timothy: ====== No, it’s not. Since the category has a terminal object (1,1), being a LCCC would imply it was cartesian closed. However, that would imply (G,X) \times — preserved the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0). ====== But it seems there is more to the story. The thing stopping the category of actions from being cartesian closed is that the category Gp of groups is not. However, the category Gpd of groupoids and the category Cat of categories are. The (2-)category Cat//’Set’ of all category actions is defined as follows: objects (A,F) are functors F : A —> Set and morphisms (f,t) : (A,F) —> (B,G) are functors f : A —> B with natural transformation t : F ==> G f. This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is ([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H g). However Cat//’Set’ is not locally cartesian closed basically because Cat is not. It is not even locally cartesian closed as a bicategory. The 2-category Gpd is cartesian closed; it is not locally cartesian closed; it is locally cartesian closed as a bicategory. Similarly, Gpd//’Set’ is locally cartesian closed as a bicategory. Often, in dealing with groups, we find groupoids help. This case is a good example and I hope helps in the applications you have in mind, Timothy. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Timothy, hi Ross, this message to highlight the importance of the construction of the category of group actions (Grp,C) over an arbirary base category C (while Timothy is just considering the case C=Sets). Indeed, the category of group actions (Grp,C) has as objects triples (G,X,\phi) consisting of a group G, an object X of C, and a monoid morphism G->C(X,X), and as maps (g,f):(G,X)->(H,Y) pairs consisting of a group morphism g:G->H and an arrow f:X->Y in C such that an obvious pentagonal diagram in Sets commutes. Composition in this category is defined by gluing together two pentagons along a commutative square. This construction seems to have quite interesting preservation properties (alas not inclucing cartesian nor local cartesian closedness). These properties may have their use in the theory of semi-abelian categories. Just a few samples: The category (Grp,Grp) of group actions in groups is equivalent to Bourn's category of split epimorphisms in Grp, also known as the category of points in Grp. The latter is known to inherit several important properties of the category Grp, such as protomodularity, regularity and exactness. In general, if C is regular (exact) then (Grp,C) as well. In the theory of cocommutative Hopf algebras over a field k, a structure thm of Gabriel-Cartier may be interpreted as saying that over an algebraically closed field of characteristic 0, the category CocommHopf_k is equivalent to (Grp,Lie_k), where Lie_k is the category of Lie k-algebras. Since the latter is exact, this shows that CocommHopf_k is exact as well. An adjunction argument implies then that CocommHopf_k is exact for any field of characteristic 0. This is the most difficult step in showing that CocommHopf_k is actually semi-abelian for any field of characterstic zero. All the best, Clemens. Le 2014-09-04 02:19, Ross Street a ??crit??:
On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:
I'm wondering whether the category of ALL group actions is locally Cartesian closed.
This is what I answered Timothy: ====== No, it???s not. Since the category has a terminal object (1,1), being a LCCC would imply it was cartesian closed. However, that would imply (G,X) \times ??? preserved the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0). ======
But it seems there is more to the story. The thing stopping the category of actions from being cartesian closed is that the category Gp of groups is not. However, the category Gpd of groupoids and the category Cat of categories are. The (2-)category Cat//???Set??? of all category actions is defined as follows: objects (A,F) are functors F : A ???> Set and morphisms (f,t) : (A,F) ???> (B,G) are functors f : A ???> B with natural transformation t : F ==> G f. This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is ([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H g).
However Cat//???Set??? is not locally cartesian closed basically because Cat is not. It is not even locally cartesian closed as a bicategory. The 2-category Gpd is cartesian closed; it is not locally cartesian closed; it is locally cartesian closed as a bicategory.
Similarly, Gpd//???Set??? is locally cartesian closed as a bicategory. Often, in dealing with groups, we find groupoids help. This case is a good example and I hope helps in the applications you have in mind, Timothy.
Ross
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
It seems that the following is in fact true: Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed. The product of (a, phi) with (b, psi) in E is of course (a x b, pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the product projections in B. The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi), ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and ev: [b,c] x b ----> c in B. This in particular applies to Cat//’Set’ as in Ross' message, seen as a fibration over Cat with reindexing along f:A--->B given by [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to pullbacks, but they don't satisfy BCC; however, right adjoints to pullback along product projections are given just by (conical) limit functors, and these do satisfy BCC. So the preceding construction applies (and a bit of fiddling about shows that this does indeed agree with Ross' prescription). As for local cartesian closure: if B is lccc, each fibre is lccc with fibrewise Pi's stable under pullback, and E--->B has all products, then it seems that each slice fibration p/A: E/A--->B/pA will satisfy the conditions in the second paragraph, whence E is also lccc. Richard On Thu, Sep 4, 2014, at 10:19 AM, Ross Street wrote:
On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:
I'm wondering whether the category of ALL group actions is locally Cartesian closed.
This is what I answered Timothy: ====== No, it’s not. Since the category has a terminal object (1,1), being a LCCC would imply it was cartesian closed. However, that would imply (G,X) \times — preserved the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0). ======
But it seems there is more to the story. The thing stopping the category of actions from being cartesian closed is that the category Gp of groups is not. However, the category Gpd of groupoids and the category Cat of categories are. The (2-)category Cat//’Set’ of all category actions is defined as follows: objects (A,F) are functors F : A —> Set and morphisms (f,t) : (A,F) —> (B,G) are functors f : A —> B with natural transformation t : F ==> G f. This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is ([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H g).
However Cat//’Set’ is not locally cartesian closed basically because Cat is not. It is not even locally cartesian closed as a bicategory. The 2-category Gpd is cartesian closed; it is not locally cartesian closed; it is locally cartesian closed as a bicategory.
Similarly, Gpd//’Set’ is locally cartesian closed as a bicategory. Often, in dealing with groups, we find groupoids help. This case is a good example and I hope helps in the applications you have in mind, Timothy.
Ross
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 2014-09-04, 10:05 PM, Richard Garner wrote:
It seems that the following is in fact true:
Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed.
The product of (a, phi) with (b, psi) in E is of course (a x b, pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the product projections in B.
The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi), ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and ev: [b,c] x b ----> c in B.
This is indeed the case and it appears as Corollary 4.12 in Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure and Applied Algebra, Volume 134, Issue 1, 5 January 1999, Pages 83-109, ISSN 0022-4049, http://dx.doi.org/10.1016/S0022-4049(97)00129-1. (http://www.sciencedirect.com/science/article/pii/S0022404997001291)
This in particular applies to Cat//'Set' as in Ross' message, seen as a fibration over Cat with reindexing along f:A--->B given by [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to pullbacks, but they don't satisfy BCC; however, right adjoints to pullback along product projections are given just by (conical) limit functors, and these do satisfy BCC. So the preceding construction applies (and a bit of fiddling about shows that this does indeed agree with Ross' prescription).
As for local cartesian closure: if B is lccc, each fibre is lccc with fibrewise Pi's stable under pullback, and E--->B has all products, then it seems that each slice fibration p/A: E/A--->B/pA will satisfy the conditions in the second paragraph, whence E is also lccc.
That is also correct, but Cat is not lccc. To get this to work, one must restrict Cat to the broad subcategory whose morphisms satisfy the Conduche condition (which is the same as exponentiability in Cat), as exposed in the nLab page http://nlab.mathforge.org/nlab/show/Conduche+functor Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 2014-09-04, 10:05 PM, Richard Garner wrote:
It seems that the following is in fact true:
Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed.
The product of (a, phi) with (b, psi) in E is of course (a x b, pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the product projections in B.
The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi), ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and ev: [b,c] x b ----> c in B.
This is indeed the case and it appears as Corollary 4.12 in Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure and Applied Algebra, Volume 134, Issue 1, 5 January 1999, Pages 83-109, ISSN 0022-4049, http://dx.doi.org/10.1016/S0022-4049(97)00129-1. (http://www.sciencedirect.com/science/article/pii/S0022404997001291)
This in particular applies to Cat//'Set' as in Ross' message, seen as a fibration over Cat with reindexing along f:A--->B given by [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to pullbacks, but they don't satisfy BCC; however, right adjoints to pullback along product projections are given just by (conical) limit functors, and these do satisfy BCC. So the preceding construction applies (and a bit of fiddling about shows that this does indeed agree with Ross' prescription).
As for local cartesian closure: if B is lccc, each fibre is lccc with fibrewise Pi's stable under pullback, and E--->B has all products, then it seems that each slice fibration p/A: E/A--->B/pA will satisfy the conditions in the second paragraph, whence E is also lccc.
That is also correct, but Cat is not lccc. To get this to work, one must restrict Cat to the broad subcategory whose morphisms satisfy the Conduche condition (which is the same as exponentiability in Cat), as exposed in the nLab page http://nlab.mathforge.org/nlab/show/Conduche+functor Claudio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Tue, 02 Sep 2014 08:28:21 PM EDT, Timothy Revell <timothy.revell@strath.ac.uk> asked:
I'm wondering whether the category of ALL group actions is locally Cartesian closed. ...
Borrowing ideas from several earlier followups to this question, I might suggest the following as a quick and easy argument for a negative answer: 1) an initial object in that "category of ALL group actions" is given by the trivial group's action on the empty set, ({e}, 0); 2) the slice of that category over ({e}, 0) "is" just the category Grp of groups and group homomorphisms (well, really, it's the full subcategory of ALL group actions on the empty set :-) but that's essentially just Grp); 3) the category of group actions is no more a LCCC than Grp is a CCC. Or have I overlooked some obvious fly in my proposed ointment :-) ? Cheers, -- Fred Linton [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On page -15 (yes, a negative page number) of the TAC reprinting of Abelian Categories (in the 2003 Forward) I wrote: The very large category [described] in Exercise 6-A -- with a few variations -- has been a great source of counterexamples over the years....In its category of abelian-group objects Ext(A,B) is a proper class iff there???s a non-zero group homomorphism from A to B (it needn???t respect the actions) hence the only injective object is the zero object (which settled a once-open problem about whether there are enough injectives in the category of abelian groups in every elementary topos with natural-numbers object. http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf On 2014-09-01 05:12, Timothy Revell wrote:
Dear All,
I'm wondering whether the category of ALL group actions is locally Cartesian closed. This is NOT the functor category [G,Set] for some category G with one object, since we allow G to vary. To be more specific the category is as follows.
- The objects are pairs (G,X), where G is a group and X is a G-Set.
- A morphism (G,X) -> (G', X') is given by a pair (h,f), where h:G->G' is a group homomorphism and f: X -> X' is a function (a morphism in Set) such that for all g in G, x in X
h(g) * f(x) = f(g * x)
where * on the left denotes the group action of G' on X' and * on the right denotes the group action of G on X.
All the best, Tim
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (8)
-
Claudio Hermida -
Clemens.BERGER＠unice.fr -
Fred E.J. Linton -
pjf -
Richard Garner -
Ross Street -
Steve Lack -
Timothy Revell